Focal plane shutters can only sync at 1/160 to 1/250th or so. This is a limitation of that type of shutter, which all DSLR and mirrorless cameras use (even if they use electronic first curtain shutters they are still limited by mechanical rear curtain). I believe the higher the sync speed the faster the curtain has to move, which will likely lead to more vibration.

Some older cameras using global electronic shutters, which we lost when we went to CMOS from CCD, can sync much faster. Like Nikon D1x and D70s.

Leaf shutter cameras can usually sync at max shutter speed, so if you really need that sort of sync speed you would have to go with a point and shoot or something. A fuji X20 can sync up to 1/2000 or maybe even 1/4000 I think.